First of all, I'd probably like to find out how its syntactic structure roughly goes, while starting to learn a brand-new language.

I learn basic sentences.( what's your name/my name is, i live in etc..) Because you can see the structure of these sentences and you learn at the same time noun and verbs. Then you can try to build new sentence with the one you learn, and add some new sentences when you need more vocabulary.

Since you are a singer, what do you think trying to remember the lyrics of songs as the beginning?

I agree with Audrey. You get a set of very useful words/verbs/structures very fast that way. You can always add more grammar theory if you like. Then, as you progress, you complete your vocabulary with the subjects you're interested in. This has also the advantage of keeping you motivated, it's harder to remember abstract structures if you don't have an immediate use for them.

When I start to learn a new language first of all I focus on the letters if it has a different alphabet and also on their pronunciation. I agree with YoshiYoshi because one of the basical things to do it to try to understand how simple phrases are built, where are the suffixes, the adjectives, the verbs and so on. In the meanwhile I listen to local radios and songs. I also write every new word I find and I never write the meaning in my language so that I am forced to make a constant effort. When I am ready to learn a bit of vocabulary I put post-its everywhere, i.e. I want to learn the word “chair” in French so I attach a post-it on all the chairs of the house where I wrote “LA CHAISE”.